Integrand size = 13, antiderivative size = 186 \[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=-\frac {x}{a}-\frac {3 \text {arctanh}(\cos (x))}{8 b}-\frac {\left (a^2-3 b^2\right ) \text {arctanh}(\cos (x))}{2 b^3}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cos (x))}{b^5}+\frac {2 \left (a^2-b^2\right )^{5/2} \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a b^5}+\frac {a \cot (x)}{b^2}+\frac {a \left (a^2-3 b^2\right ) \cot (x)}{b^4}+\frac {a \cot ^3(x)}{3 b^2}-\frac {3 \cot (x) \csc (x)}{8 b}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}-\frac {\cot (x) \csc ^3(x)}{4 b} \]
-x/a-3/8*arctanh(cos(x))/b-1/2*(a^2-3*b^2)*arctanh(cos(x))/b^3-(a^4-3*a^2* b^2+3*b^4)*arctanh(cos(x))/b^5+2*(a^2-b^2)^(5/2)*arctanh((a+b*tan(1/2*x))/ (a^2-b^2)^(1/2))/a/b^5+a*cot(x)/b^2+a*(a^2-3*b^2)*cot(x)/b^4+1/3*a*cot(x)^ 3/b^2-3/8*cot(x)*csc(x)/b-1/2*(a^2-3*b^2)*cot(x)*csc(x)/b^3-1/4*cot(x)*csc (x)^3/b
Time = 1.10 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=\frac {-192 b^5 x+384 \left (-a^2+b^2\right )^{5/2} \arctan \left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )+32 a^2 b \left (3 a^2-7 b^2\right ) \cot \left (\frac {x}{2}\right )-24 a^3 b^2 \csc ^2\left (\frac {x}{2}\right )+54 a b^4 \csc ^2\left (\frac {x}{2}\right )-3 a b^4 \csc ^4\left (\frac {x}{2}\right )-192 a^5 \log \left (\cos \left (\frac {x}{2}\right )\right )+480 a^3 b^2 \log \left (\cos \left (\frac {x}{2}\right )\right )-360 a b^4 \log \left (\cos \left (\frac {x}{2}\right )\right )+192 a^5 \log \left (\sin \left (\frac {x}{2}\right )\right )-480 a^3 b^2 \log \left (\sin \left (\frac {x}{2}\right )\right )+360 a b^4 \log \left (\sin \left (\frac {x}{2}\right )\right )+24 a^3 b^2 \sec ^2\left (\frac {x}{2}\right )-54 a b^4 \sec ^2\left (\frac {x}{2}\right )+3 a b^4 \sec ^4\left (\frac {x}{2}\right )-64 a^2 b^3 \csc ^3(x) \sin ^4\left (\frac {x}{2}\right )+4 a^2 b^3 \csc ^4\left (\frac {x}{2}\right ) \sin (x)-96 a^4 b \tan \left (\frac {x}{2}\right )+224 a^2 b^3 \tan \left (\frac {x}{2}\right )}{192 a b^5} \]
(-192*b^5*x + 384*(-a^2 + b^2)^(5/2)*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b ^2]] + 32*a^2*b*(3*a^2 - 7*b^2)*Cot[x/2] - 24*a^3*b^2*Csc[x/2]^2 + 54*a*b^ 4*Csc[x/2]^2 - 3*a*b^4*Csc[x/2]^4 - 192*a^5*Log[Cos[x/2]] + 480*a^3*b^2*Lo g[Cos[x/2]] - 360*a*b^4*Log[Cos[x/2]] + 192*a^5*Log[Sin[x/2]] - 480*a^3*b^ 2*Log[Sin[x/2]] + 360*a*b^4*Log[Sin[x/2]] + 24*a^3*b^2*Sec[x/2]^2 - 54*a*b ^4*Sec[x/2]^2 + 3*a*b^4*Sec[x/2]^4 - 64*a^2*b^3*Csc[x]^3*Sin[x/2]^4 + 4*a^ 2*b^3*Csc[x/2]^4*Sin[x] - 96*a^4*b*Tan[x/2] + 224*a^2*b^3*Tan[x/2])/(192*a *b^5)
Time = 0.54 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4386, 3042, 3376, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (x)^6}{a+b \csc (x)}dx\) |
| \(\Big \downarrow \) 4386 |
| \(\displaystyle \int \frac {\cos (x) \cot ^5(x)}{a \sin (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (x)^6}{\sin (x)^5 (a \sin (x)+b)}dx\) |
| \(\Big \downarrow \) 3376 |
| \(\displaystyle \int \left (\frac {\left (3 a b^2-a^3\right ) \csc ^2(x)}{b^4}-\frac {\left (a^2-b^2\right )^3}{a b^5 (a \sin (x)+b)}+\frac {\left (a^2-3 b^2\right ) \csc ^3(x)}{b^3}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \csc (x)}{b^5}-\frac {a \csc ^4(x)}{b^2}-\frac {1}{a}+\frac {\csc ^5(x)}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (a^2-b^2\right )^{5/2} \text {arctanh}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a b^5}-\frac {\left (a^2-3 b^2\right ) \text {arctanh}(\cos (x))}{2 b^3}+\frac {a \left (a^2-3 b^2\right ) \cot (x)}{b^4}-\frac {\left (a^2-3 b^2\right ) \cot (x) \csc (x)}{2 b^3}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cos (x))}{b^5}+\frac {a \cot ^3(x)}{3 b^2}+\frac {a \cot (x)}{b^2}-\frac {x}{a}-\frac {3 \text {arctanh}(\cos (x))}{8 b}-\frac {\cot (x) \csc ^3(x)}{4 b}-\frac {3 \cot (x) \csc (x)}{8 b}\) |
-(x/a) - (3*ArcTanh[Cos[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTanh[Cos[x]])/(2*b^ 3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTanh[Cos[x]])/b^5 + (2*(a^2 - b^2)^(5/2 )*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b^5) + (a*Cot[x])/b^2 + (a *(a^2 - 3*b^2)*Cot[x])/b^4 + (a*Cot[x]^3)/(3*b^2) - (3*Cot[x]*Csc[x])/(8*b ) - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/(2*b^3) - (Cot[x]*Csc[x]^3)/(4*b)
3.1.23.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(d*sin[ e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; Fr eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && ( LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Int[Cos[c + d*x]^m*((b + a*Sin[c + d*x])^n/Sin[c + d*x]^(m + n)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])
Time = 3.31 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(-\frac {-\frac {b^{3} \tan \left (\frac {x}{2}\right )^{4}}{4}+\frac {2 \tan \left (\frac {x}{2}\right )^{3} a \,b^{2}}{3}-2 \tan \left (\frac {x}{2}\right )^{2} a^{2} b +4 \tan \left (\frac {x}{2}\right )^{2} b^{3}+8 \tan \left (\frac {x}{2}\right ) a^{3}-18 \tan \left (\frac {x}{2}\right ) a \,b^{2}}{16 b^{4}}-\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}-\frac {1}{64 b \tan \left (\frac {x}{2}\right )^{4}}-\frac {4 a^{2}-8 b^{2}}{32 b^{3} \tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{4}-40 a^{2} b^{2}+30 b^{4}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16 b^{5}}+\frac {a}{24 b^{2} \tan \left (\frac {x}{2}\right )^{3}}+\frac {a \left (4 a^{2}-9 b^{2}\right )}{8 b^{4} \tan \left (\frac {x}{2}\right )}+\frac {\left (-32 a^{6}+96 a^{4} b^{2}-96 a^{2} b^{4}+32 b^{6}\right ) \arctan \left (\frac {2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{16 a \,b^{5} \sqrt {-a^{2}+b^{2}}}\) | \(251\) |
| risch | \(-\frac {x}{a}-\frac {-168 i a \,b^{2} {\mathrm e}^{4 i x}+72 i a^{3} {\mathrm e}^{4 i x}-12 a^{2} b \,{\mathrm e}^{7 i x}+27 b^{3} {\mathrm e}^{7 i x}+24 i a^{3}-56 i a \,b^{2}+12 a^{2} b \,{\mathrm e}^{5 i x}-3 b^{3} {\mathrm e}^{5 i x}+72 i a \,b^{2} {\mathrm e}^{6 i x}-72 i a^{3} {\mathrm e}^{2 i x}+12 a^{2} b \,{\mathrm e}^{3 i x}-3 b^{3} {\mathrm e}^{3 i x}+152 i a \,b^{2} {\mathrm e}^{2 i x}-24 i a^{3} {\mathrm e}^{6 i x}-12 a^{2} b \,{\mathrm e}^{i x}+27 \,{\mathrm e}^{i x} b^{3}}{12 b^{4} \left ({\mathrm e}^{2 i x}-1\right )^{4}}-\frac {\sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i b -\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{5}}+\frac {2 \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i x}+\frac {i b -\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{3}}-\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {i b -\sqrt {a^{2}-b^{2}}}{a}\right )}{b a}+\frac {\sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i b +\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{5}}-\frac {2 \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i x}+\frac {i b +\sqrt {a^{2}-b^{2}}}{a}\right )}{b^{3}}+\frac {\sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {i b +\sqrt {a^{2}-b^{2}}}{a}\right )}{b a}-\frac {\ln \left ({\mathrm e}^{i x}+1\right ) a^{4}}{b^{5}}+\frac {5 \ln \left ({\mathrm e}^{i x}+1\right ) a^{2}}{2 b^{3}}-\frac {15 \ln \left ({\mathrm e}^{i x}+1\right )}{8 b}+\frac {\ln \left ({\mathrm e}^{i x}-1\right ) a^{4}}{b^{5}}-\frac {5 \ln \left ({\mathrm e}^{i x}-1\right ) a^{2}}{2 b^{3}}+\frac {15 \ln \left ({\mathrm e}^{i x}-1\right )}{8 b}\) | \(554\) |
-1/16/b^4*(-1/4*b^3*tan(1/2*x)^4+2/3*tan(1/2*x)^3*a*b^2-2*tan(1/2*x)^2*a^2 *b+4*tan(1/2*x)^2*b^3+8*tan(1/2*x)*a^3-18*tan(1/2*x)*a*b^2)-2/a*arctan(tan (1/2*x))-1/64/b/tan(1/2*x)^4-1/32*(4*a^2-8*b^2)/b^3/tan(1/2*x)^2+1/16/b^5* (16*a^4-40*a^2*b^2+30*b^4)*ln(tan(1/2*x))+1/24*a/b^2/tan(1/2*x)^3+1/8*a*(4 *a^2-9*b^2)/b^4/tan(1/2*x)+1/16*(-32*a^6+96*a^4*b^2-96*a^2*b^4+32*b^6)/a/b ^5/(-a^2+b^2)^(1/2)*arctan(1/2*(2*b*tan(1/2*x)+2*a)/(-a^2+b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (168) = 336\).
Time = 0.57 (sec) , antiderivative size = 852, normalized size of antiderivative = 4.58 \[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=\text {Too large to display} \]
[-1/48*(48*b^5*x*cos(x)^4 - 96*b^5*x*cos(x)^2 + 48*b^5*x - 6*(4*a^3*b^2 - 9*a*b^4)*cos(x)^3 - 24*((a^4 - 2*a^2*b^2 + b^4)*cos(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)*sqrt(a^2 - b^2)*log(((a^2 - 2 *b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x)) *sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 6*(4*a^3*b^ 2 - 7*a*b^4)*cos(x) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b ^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log( 1/2*cos(x) + 1/2) - 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(-1 /2*cos(x) + 1/2) + 16*((3*a^4*b - 7*a^2*b^3)*cos(x)^3 - 3*(a^4*b - 2*a^2*b ^3)*cos(x))*sin(x))/(a*b^5*cos(x)^4 - 2*a*b^5*cos(x)^2 + a*b^5), -1/48*(48 *b^5*x*cos(x)^4 - 96*b^5*x*cos(x)^2 + 48*b^5*x - 6*(4*a^3*b^2 - 9*a*b^4)*c os(x)^3 - 48*((a^4 - 2*a^2*b^2 + b^4)*cos(x)^4 + a^4 - 2*a^2*b^2 + b^4 - 2 *(a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^ 2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) + 6*(4*a^3*b^2 - 7*a*b^4)*cos(x) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x) ^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) - 3 *(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^4 - 2*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2) + 16 *((3*a^4*b - 7*a^2*b^3)*cos(x)^3 - 3*(a^4*b - 2*a^2*b^3)*cos(x))*sin(x)...
\[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=\int \frac {\cot ^{6}{\left (x \right )}}{a + b \csc {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.29 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=-\frac {x}{a} + \frac {3 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{4} - 8 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} - 48 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 96 \, a^{3} \tan \left (\frac {1}{2} \, x\right ) + 216 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )}{192 \, b^{4}} + \frac {{\left (8 \, a^{4} - 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{8 \, b^{5}} - \frac {2 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a b^{5}} - \frac {400 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{4} - 1000 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 750 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} - 96 \, a^{3} b \tan \left (\frac {1}{2} \, x\right )^{3} + 216 \, a b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 48 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a b^{3} \tan \left (\frac {1}{2} \, x\right ) + 3 \, b^{4}}{192 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{4}} \]
-x/a + 1/192*(3*b^3*tan(1/2*x)^4 - 8*a*b^2*tan(1/2*x)^3 + 24*a^2*b*tan(1/2 *x)^2 - 48*b^3*tan(1/2*x)^2 - 96*a^3*tan(1/2*x) + 216*a*b^2*tan(1/2*x))/b^ 4 + 1/8*(8*a^4 - 20*a^2*b^2 + 15*b^4)*log(abs(tan(1/2*x)))/b^5 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b* tan(1/2*x) + a)/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a*b^5) - 1/192*(400*a ^4*tan(1/2*x)^4 - 1000*a^2*b^2*tan(1/2*x)^4 + 750*b^4*tan(1/2*x)^4 - 96*a^ 3*b*tan(1/2*x)^3 + 216*a*b^3*tan(1/2*x)^3 + 24*a^2*b^2*tan(1/2*x)^2 - 48*b ^4*tan(1/2*x)^2 - 8*a*b^3*tan(1/2*x) + 3*b^4)/(b^5*tan(1/2*x)^4)
Time = 20.28 (sec) , antiderivative size = 4075, normalized size of antiderivative = 21.91 \[ \int \frac {\cot ^6(x)}{a+b \csc (x)} \, dx=\text {Too large to display} \]
tan(x/2)*(a/(8*b^2) + (2*a*(1/(2*b) - a^2/(4*b^3)))/b) - tan(x/2)^2*(1/(4* b) - a^2/(8*b^3)) + tan(x/2)^4/(64*b) - (2*atan((((((4*(24*a*b^16 + 58*a^3 *b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440*a^9*b^8 + 184*a^11*b^6 - 32*a^13 *b^4))/b^12 + (((((4*(24*a^3*b^16 - 32*a^5*b^14))/b^12 - (4*tan(x/2)*(16*a ^2*b^17 - 136*a^4*b^15 + 128*a^6*b^13))/b^12)*1i)/a - (4*(93*a^3*b^15 - 24 0*a^5*b^13 + 208*a^7*b^11 - 64*a^9*b^9))/b^12 + (4*tan(x/2)*(62*a^2*b^16 - 335*a^4*b^14 + 604*a^6*b^12 - 456*a^8*b^10 + 128*a^10*b^8))/b^12)*1i)/a - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 252*a^4*b^13 - 148*a^6*b^11 + 100*a ^8*b^9 - 40*a^10*b^7 + 8*a^12*b^5))/b^12)*1i)/a - (4*(53*a*b^15 - 120*a^3* b^13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5))/b^12 + (4*tan( x/2)*(8*a^16 + 62*b^16 - 410*a^2*b^14 + 929*a^4*b^12 - 1096*a^6*b^10 + 873 *a^8*b^8 - 550*a^10*b^6 + 255*a^12*b^4 - 68*a^14*b^2))/b^12)/a - ((4*(53*a *b^15 - 120*a^3*b^13 + 48*a^5*b^11 + 56*a^7*b^9 - 48*a^9*b^7 + 8*a^11*b^5) )/b^12 + (((4*(24*a*b^16 + 58*a^3*b^14 - 345*a^5*b^12 + 543*a^7*b^10 - 440 *a^9*b^8 + 184*a^11*b^6 - 32*a^13*b^4))/b^12 + (((4*(93*a^3*b^15 - 240*a^5 *b^13 + 208*a^7*b^11 - 64*a^9*b^9))/b^12 + (((4*(24*a^3*b^16 - 32*a^5*b^14 ))/b^12 - (4*tan(x/2)*(16*a^2*b^17 - 136*a^4*b^15 + 128*a^6*b^13))/b^12)*1 i)/a - (4*tan(x/2)*(62*a^2*b^16 - 335*a^4*b^14 + 604*a^6*b^12 - 456*a^8*b^ 10 + 128*a^10*b^8))/b^12)*1i)/a - (4*tan(x/2)*(16*b^17 - 180*a^2*b^15 + 25 2*a^4*b^13 - 148*a^6*b^11 + 100*a^8*b^9 - 40*a^10*b^7 + 8*a^12*b^5))/b^...